Square Root Calculator
Our free arithmetic calculator solves square root problems. Get worked examples, visual aids, and downloadable results.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
sqrt(n) = n^(1/2)
The square root of n is the value that, when multiplied by itself, gives n. For non-perfect squares, the simplified radical form extracts the largest perfect square factor: sqrt(a*b^2) = b*sqrt(a).
Worked Examples
Example 1: Square Root of 144
Problem:Find the square root of 144 and determine if it is a perfect square.
Solution:Check if 144 is a perfect square: 12 x 12 = 144. Yes, it is a perfect square.\nThe square root of 144 = 12 (exact integer result).\nVerification: 12^2 = 144.\nPrime factorization: 144 = 2^4 x 3^2 = (2^2 x 3)^2 = 12^2.\nThis confirms 144 is a perfect square with root 12.
Result:sqrt(144) = 12 (perfect square)
Example 2: Square Root of 50 Simplified
Problem:Simplify the square root of 50 into radical form and find its decimal value.
Solution:Factor 50 to find perfect square factors: 50 = 25 x 2.\n25 is a perfect square (5^2).\nsqrt(50) = sqrt(25 x 2) = sqrt(25) x sqrt(2) = 5 x sqrt(2).\nDecimal value: 5 x 1.41421356 = 7.07106781.\nVerification: 7.07106781^2 = 49.99999... which is approximately 50.
Result:sqrt(50) = 5 * sqrt(2) = 7.07106781
Frequently Asked Questions
What is a square root and how is it defined?
A square root of a number is a value that, when multiplied by itself, produces the original number. The square root of 25 is 5 because 5 times 5 equals 25. Every positive number actually has two square roots: a positive root (called the principal square root) and a negative root. For 25, both +5 and -5 are square roots since (-5) times (-5) also equals 25. The radical symbol always refers to the principal (positive) root. Square roots are the inverse operation of squaring a number. This concept extends throughout mathematics, appearing in the Pythagorean theorem, quadratic formula, standard deviation calculations, and many areas of science and engineering.
What is a perfect square and how do I recognize one?
A perfect square is a number that results from multiplying an integer by itself. The first twenty perfect squares are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, and 400. You can recognize perfect squares by checking if their square root is a whole number. Perfect squares have several useful properties: they always end in 0, 1, 4, 5, 6, or 9 (never 2, 3, 7, or 8). Also, the digital root of a perfect square is always 1, 4, 7, or 9. Memorizing perfect squares up to at least 400 is helpful for mental math, standardized tests, and quickly simplifying radical expressions in algebra.
How does the Newton-Raphson method calculate square roots?
The Newton-Raphson method (also called the Babylonian method for square roots) is an iterative algorithm that converges extremely quickly to the true square root value. Start with an initial guess, then repeatedly apply the formula: new guess equals (guess + number/guess) divided by 2. Each iteration roughly doubles the number of correct digits. For example, finding the square root of 50: start with guess 25, then (25 + 50/25)/2 = 13.5, then (13.5 + 50/13.5)/2 = 8.602, then (8.602 + 50/8.602)/2 = 7.208, then quickly converging to 7.07107. After just 5-6 iterations, you typically have 10+ correct decimal places, making this method extremely efficient for computation.
What is the square root of a negative number?
The square root of a negative number does not exist in the real number system because no real number multiplied by itself can produce a negative result. Positive times positive is positive, and negative times negative is also positive. To handle this, mathematicians created the imaginary unit i, defined as the square root of negative one. Using i, the square root of any negative number can be expressed: the square root of -9 equals 3i, the square root of -2 equals i times the square root of 2. These imaginary numbers are not just abstract concepts. They are essential in electrical engineering for AC circuit analysis, quantum mechanics, signal processing, and control systems engineering.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy